This invention relates to a method of predicting or calculating physical quantities of a fluid or a magnetofluid which are utilized in wind channel (air duct) experiments during product development of automobiles and airplanes, or in the control of injection molding processes of plastic resins.
A physical quantity f of a fluid or a magnetofluid which propagates through space at a velocity u is governed by the fundamental equation: EQU .differential.f/.differential.t+(u.multidot..gradient.)f=0 (1)
where t is the time variable, and .gradient. is the nabla vector (.differential./.differential.x, .differential./.differential.y, .differential./.differential.z). During product development of automobiles or airplanes, or in injection molding of plastics, it often becomes necessary to solve the above equation numerically and predict the time evolution of the spatial profile of a physical quantity f of a fluid or a magnetofluid from an initial spatial profile thereof. In the case where the velocity u is constant, the equation (1) can be solved analytically and there is no need for a numerical method of solution. However, when the velocity u is a function of the space variable (u=u(x,y,z)) and thus varies over the space, or is a function of space and time (u=u(x,y,z,t)) and thus varies over the space and time, there is no known general method of obtaining an analytic solution and hence it becomes necessary to solve the equation (1) approximately by means of a numerical method.
One such numerical method, known as the cubic-interpolated pseudoparticle (CIP) method, is disclosed, for example, in T. Yabe and E. Takei: "A New Higher-Order Godunov Method for General Hyperbolic Equations", Journal of Physical Society of Japan, Vol. 57, No. 8, August 1988, pp. 2598-2601, the disclosure of which article is incorporated herein by reference. According to the CIP method, a grid or mesh of a predetermined mesh spacing (.DELTA.x and .DELTA.t) is set over the space and time, and the time evolution of the spatial profile of the physical quantity f is calculated numerically from the initial profile. At each step of repeated calculations, each evolution of the spatial profile of the physical quantity f over a very short time .DELTA.t is calculated. By repeating a predetermined number of such calculations, the spatial profile of the physical quantity f after a predetermined interval of time can be obtained.
FIG. 1 shows a numerical solution (represented by a dotted curve) of one-dimensional non-attenuating triangular wave motion obtained by the CIP method disclosed in the above-mentioned article. The solution has been obtained after 1000 steps of repeated calculations of the time evolutions of the spatial profile of a triangular wave, each time step corresponding to the time increment .DELTA.t. For comparison, the solid curve represents the analytic or the true solution. The calculated or predicted height of the triangular wave profile is 0.91584, as opposed to the true height 1.0000 of the analytic solution. Thus, the prediction contains an error of about 8%. This result shows the fact that prediction precision suffers deterioration due to the numerical viscosity.
The generally adopted measure for enhancing the prediction precision of the numerical solution has been to reduce the mesh spacings and to increase the number of mesh points of the mesh or grid.
The above numerical method of solving the fundamental equation has the following disadvantage. If the number of mesh points is not large enough, the prediction precision tends to deteriorate in the region where the physical quantity f or the inclination thereof (.differential.f/.differential.x, .differential.f/.differential.y, .differential.f/.differential.z) varies abruptly and hence where the prediction errors tend to be generated due to the numerical viscosity. On the other hand, if the number of mesh points are increased to guarantee a precise solution, the time required for the calculation increases unduly.